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Maths

Congruent and similar shapes

Similar triangles

We know that two shapes are similar if their corresponding angles are equal, and their corresponding sides in the same proportion. The same is true for similar triangles.

To prove that two triangles are similar, we have to show that one (not all) of the following statements is true:

1. The three sides are in the same proportion.

image: two triangle: left triangle: left-side: 3cm, right-side: 5cm, bottom: 4cm. Right triangle: left-side: 4.5cm, right-side: 7.5cm, bottom: 6cm.

2. Two sides are in the same proportion, and their included angle is equal.

image: two triangles: left triangle: right-side: 5cm, bottom-right corner: 30 degree, bottom: 3cm. Right triangle: right-side: 10cm, bottom-right corner: 30 degree, bottom: 6cm

3. The three angles of the first triangle are equal to the three angles of the second triangle.

image: two triangles: left triangle: top Y corner: 85 degrees, right Z corner: 40 degrees, left corner: X. Right triangle: same labels: Y: 85 degrees, X: 55 degrees.

We have only shown two angles in the above triangles. If we calculate the missing angles, we find that angle ZXY is 55° and angle YZX is 40°. Therefore, all corresponding angles are equal.

Question

Are triangles PQR and SQT similar? If so, which of the above statements applies?

image: triangle: top corners: Q, right: T, right corner: R, left corner: P, left: S. Lengths: QT: 3cm, TR: 5cm, PS: 2cm, SQ: 4cm.

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Answer

This question is much easier if we draw the triangles separately.

image: two triangle: left triangle: top: Q, right: T, left: S. Lengths: QT: 3cm, SQ: 4cms. Right triangle: same labels: QR: 8cm, PQ: 6cm.

QR corresponds to QS (QR = 2QS) and QP corresponds to QT (QP = 2QT). Angle Q is the same in both triangles. Therefore, they are similar, because two sides are in the same proportion and their included angle is equal (statement 2).

Notice that they are similar, even though they are 'mirror' images (of different sizes).

Now try this one.

Question

Are triangles ABC and DBE similar? If so, which of the above statements (1, 2 or 3) applies?

triangle, from clockwise directions lengths from right: labels BE:4cm, labels EC: 2cms, bottom: labels CA: 5cm. left side have two labels, AD: opposite to EC through to the top B,

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Answer

ABC and DBE are similar, because all of the corresponding angles are equal (statement 3).

If you found this difficult, remember that angle B is common to both triangles. The rules of parallel lines state that

BDE = BAC (corresponding angles)

and that BED = BCA (corresponding angles).

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